Technical note: The Weibull distribution as an extreme value alternative for annual maxima

Abstract. The generalized extreme value distribution (GEV) for largest extremes is widely applied to single-site annual maxima sequences for estimating exceedance probabilities for application to design magnitudes under conditions of stationarity. However, the GEV is not the only mode for application of classical extreme value theory to recorded maxima. An alternative approach is to apply specific transformations to the maxima, enabling different but equivalent exceedance statements. For example, the probability that annual flood maxima will exceed some magnitude e is the same as the probability that reciprocals of the maxima will be less than 1/ε. The transformed maxima considered here represent sample minima, where the sample is the number of transformed independent individual events per year. For sufficiently large sample sizes, this leads to just one of the extreme value distributions for design purposes – the Weibull distribution for minima. This extreme value distribution arises because it is the limit stable expression for describing distributions of large-sample minima when a lower bound is present.

There is no way of telling whether a good Weibull fit to transformed annual maxima indicates that sample sizes are sufficiently large for the Weibull extreme value approximation to apply. It could happen that a good fit is simply a fortuitous empirical matching to data from transformation selection. However, a similar issue also applies to the GEV which is itself a flexible distribution capable of empirical matching to data.

It is not possible to make a case for the Weibull distribution by application to a range of annual maxima because any number of different transformations might be applied to achieve good Weibull fits. Instead, two simple synthetic examples are used to illustrate how a good fit to annual maxima by the GEV could lead to an incorrect conclusion, in contrast to the Weibull approximation applied to the same examples.